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Convergence Rates for the Quantum Central Limit Theorem

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Convergence Rates for the Quantum Central Limit Theorem Commun. Math. Phys. 383, 223–279 (2021) Communications in Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-021-03988-1 Mathematical Physics Convergence Rates for the Quantum Central Limit Theorem Simon Becker1 , Nilanjana Datta1, Ludovico Lami2,3, Cambyse Rouzé4 1 Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences University of Cambridge, Cambridge CB3 0WA,UK. E-mail: [email protected]; [email protected] 2 School of Mathematical Sciences and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, University Park, Nottingham NG7 2RD, UK. 3 Institute of Theoretical Physics and IQST, Universität Ulm, Albert-Einstein-Allee, 89069 Ulm, Germany. E-mail: [email protected] 4 Zentrum Mathematik, Technische Universität München, 85748 Garching, Germany. E-mail: [email protected] Received: 20 February 2020 / Accepted: 15 January 2021 Published online: 15 February 2021 – © The Author(s) 2021 Contents 1. Introduction ................................. 224 2. Notation and Definitions ........................... 228 2.1 Mathematical notation ......................... 228 2.2 Definitions ................................ 229 2.2.1 Quantum information with continuous variables .......... 229 2.2.2 Phase space formalism ....................... 231 2.3 Moments ................................ 233 2.4 Quantum convolution .......................... 235 3. Cushen and Hudson’s Quantum Central Limit Theorem .......... 236 4. Main Results ................................. 237 4.1 Quantitative bounds in the QCLT .................... 237 4.2 Optimality of convergence rates and necessity of finite second moments in the QCLT ............................... 239 4.3 Applications to capacity of cascades of beam splitters with non-Gaussian environment ............................... 240 4.4 New results on quantum characteristic functions ............ 242 5. New Results on Quantum Characteristic Functions: Proofs ......... 242 5.1 Quantum–Classical Correspondence .................. 242 5.2 Decay estimates on the quantum characteristic function ........ 244 6. Quantitative Bounds in the QCLT: Proofs .................. 247 6.1 Preliminary steps ............................ 248 6.1.1 Williamson form .......................... 248 6.1.2 Local-tail decomposition ...................... 248 6.2 Proofs of convergence rates in Hilbert–Schmidt distance ....... 251 6.3 Convergence in trace distance and relative entropy ........... 254 224 S. Becker, N. Datta, L. Lami, C. Rouzé 7. Optimality of Convergence Rates and Necessity of Finite Second Moments in the QCLT: Proofs .............................. 255 7.1 Failure of convergence for states with unbounded energy ....... 255 7.2 Optimality of the convergence rates ................... 257 8. Cascade of Beam Splitters: Proofs ...................... 262 8.1 Generalities of the cascade channels .................. 262 8.2 On the effective environment state ................... 263 8.3 Approximating cascade channels .................... 265 Acknowledgements ................................ 268 Appendix A: Standard Moments Versus Phase Space Moments: The Integer Case 268 Abstract: Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n-splitter, which is fed with n copies of a centred state ρ with finite second moments, converges to the Gaussian state with the same first and second moments as ρ. Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate O n−1/2 in the Hilbert–Schmidt norm whenever the third moments of ρ are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m-mode state enters a cascade of n beam splitters of equal transmissivities λ1/n fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, − 1 witharateO n 2(m+1) . This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function χρ is uniformly bounded by some ηρ < 1 outside of any neighbourhood of the origin; also, ηρ can be made to depend only on the energy of the state ρ. 1. Introduction The Central Limit Theorem (CLT) is one of the cornerstones of probability theory. This theorem and its various extensions have found numerous applications in diverse fields including mathematics, physics, information theory, economics and psychology. Any limit theorem becomes more valuable if it is accompanied by estimates for rates of con- vergence. The Berry–Esseen theorem (see e.g. [1]), which gives the rate of convergence of the distribution of the scaled sum of independent and identically distributed (i.i.d.) random variables to a normal distribution, thus provides an important refinement of the CLT. Convergence Rates for the Quantum Central Limit Theorem 225 The first results on quantum analogues of the CLT were obtained in the early 1970s by Cushen and Hudson [2], and Hepp and Lieb [3,4]. The approach of [3] was generalised by Giri and von Waldenfels [5] a few years later. These papers were followed by numerous other quantum versions of the CLT in the context of quantum statistical mechanics [6– 14], quantum field theory [15–17], von Neumann algebras [18,19], free probability [20], noncommutative stochastic processes [21] and quantum information theory [22–24]. For a more detailed list of papers on noncommutative or quantum central limit theorems (QCLT), see for example [19,25] and references therein. A partially quantitative central limit theorem for unsharp measurements has been obtained in [26]. An important pair of non-commuting observables is the pair (x, p) of canonically conjugate operators, which obey Heisenberg’s canonical commutation relations (CCR) [x, p]=iI, where I denotes the identity operator.1 These observables could be, for example, the position and momentum operators of a quantum particle, or the so-called position and momentum quadratures of a single-mode bosonic field, described in the . 2 quantum mechanical picture by the Hilbert space H1 .= L (R) – the space of square integrable functions on R. The√ corresponding annihilation√ and creation operators are constructed as a .= (x + ip)/ 2 and a† .= (x − ip)/ 2. When expressed in terms of a, a†, the CCR take the form [a, a†]=I . Quantum states are represented by density operators, i.e. positive semi-definite trace class operators with unit trace. A state ρ of a continuous variable quantum system is uniquely identified by its characteristic function, defined for all z ∈ C by χρ(z) .= † ∗ Tr ρ eza −z a . The special class of Gaussian states comprises all quantum states whose characteristic function is the (classical) characteristic function of a normal random vari- able on C.2 Exactly as in the classical case, a quantum Gaussian state is uniquely defined by its mean and covariance matrix. Cushen and Hudson [2] proved a quantum CLT for a sequence of pairs of such canonically conjugate operators {(xn, pn) : n = 1, 2,...}, with each pair acting on a distinct copy of the Hilbert space H1. More precisely, they showed that sequences that are stochastically independent and identically distributed, and have finite covariance matrix and zero mean with respect to a quantum state ρ (given by a density operator on H1), are such that their scaled sums converge in distribution to a normal limit distribution [2, Theorem 1]. Their result admits a physical interpretation in terms of a passive quantum op- tical element known as the n-splitter. This can be thought of as the unitary opera- tor Un-split that acts on n annihilation operators of n independent optical modes as π † 2 jk = .= n i Un-split a j Un-split k Fjkak, where Fjk e is the discrete Fourier transform matrix. Passivity here means that U commutes with the canonical Hamiltonian of n-split , † = ρ the field, i.e. Un-split j a j a j 0. When n identical copies of a state are combined by means of an n-splitter, and all but the first output modes are traced away, the resulting output state is called the n-fold quantum convolution of ρ, and denoted by ρn.This nomenclature is justified by the fact that the characteristic function χρ σ of two states ρ and σ is equal to the product of the characteristic functions of ρ and σ , a relation anal- ogous to that satisfied by characteristic functions of convolutions of classical random variables. Observe state ρn can also be obtained as the output of a cascade of n − 1 1 Throughout this paper we set = 1. ∗ ∗ 2 . zX −z X The characteristic function of a complex-valued random variable X is defined by χX (z) .= E e . 226 S. Becker, N. Datta, L. Lami, C. Rouzé beam splitters with suitably tuned transmissivities λ j = j/( j +1) for j = 1, 2,...n −1 (see Fig. 1a). Cushen and Hudson’s result is that if ρ is a centred state (i.e. with zero mean) and has n finite second moments, its convolutions ρ converge to the Gaussian state ρG with the same first and second moments as ρ in the limit n →∞(Theorem 3). In [2, Theorem 1], the convergence is with respect to the weak topology of the Banach space of trace class operators, which translates to pointwise convergence of the corresponding characteristic functions, by a quantum analogue of Levy’s lemma that is also proven in [2].
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